Degenerations of cluster type varieties

TL;DR

The paper analyzes degenerations of cluster type varieties and pairs, establishing that toric degenerations are finite quotients of toric models and that, under suitable hypotheses, cluster type degenerations are finite quotients of cluster type models. It develops a framework using finite covers, complements, and dual complexes to control central fibers and boundary divisors, and proves a general existence result for 1-complements in degenerations of klt surfaces. A key outcome is that almost all weighted projective planes $\P(a,b,c)$ lack non-trivial $ ext{Q}$-Gorenstein klt degenerations, with a corollary for Markov triples giving no non-trivial degenerations of $\P(a^2,b^2,c^2)$. The work also yields a complete classification of degenerations of $\P(1,1,n)$ for $n\ge 3$ and raises questions about higher-dimensional cluster type degenerations and iteration of degenerations.

Abstract

We study degenerations of cluster type varieties and pairs. Our first theorem proves that degenerations of toric pairs are finite quotients of toric pairs. In a similar vein, under some mild conditions, we prove that degenerations of cluster type pairs are finite quotients of cluster type pairs. Then, we focus on degenerations of cluster type surfaces. We give some general criteria for the existence of $1$-complements on degenerations of toric surfaces. We prove that for almost all $(a,b,c)\in \mathbb{Z}_{\geq 1}^3$ the weighted projective plane $\mathbb{P}(a,b,c)$ has no non-trivial degenerations. In particular, for a Markov triple $(a,b,c)\in \mathbb{Z}_{\geq 2}^3$, we prove that $\mathbb{P}(a^2,b^2,c^2)$ admits no non-trivial degenerations. Finally, we give a complete classification of the degenerations of $\mathbb{P}(1,1,n)$ for $n\geq 3$.

Theorems & Definitions (59)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Definition 2.4
• Definition 2.5